Source-linked AI summary
Neural Renormalization Group Flow for Percolation
Anaclara Alvez, Luca Camagna, Sergio Chibbaro, Cyril Furtlehner, François Landes, Gianluca Manzan, Lorenzo Mensi
TL;DR
The paper examines a scale-shared encoder–decoder for predicting percolation and reconstructing largest-cluster masks across lattice scales. The same local rules are reused at every scale, so the trainable parameter count remains independent of lattice size.
Problem
The paper examines whether one scale-shared encoder–decoder can predict percolation and largest-cluster membership across lattice sizes.
Method
A recursively applied shared encoder maps configurations to a latent vector for crossing-probability prediction, while a shared decoder reconstructs largest-cluster masks.
Results
The architecture uses the same number of trainable parameters for all lattice sizes.
Takeaways & Limitations
Scale sharing makes the model’s parameter count independent of lattice size.
Abstract
from arXiv · showhide
Machine learning offers a possible route to data-driven real-space renormalization when the relevant observables are nonlocal and difficult to prescribe explicitly. We explore this idea for two-dimensional site percolation developping a supervised, scale-shared neural architecture. The model recursively applies the same learned coarse-graining rule across scales, producing a latent field from which the crossing probability is predicted, while a corresponding fine-graining decoder reconstructs the largest-cluster mask. Trained only on small lattices, the model extrapolates to substantially larger systems, recovers the spanning cluster with high fidelity, and produces observables obeying the expected finite-size scaling near the critical point. We observe that to get such performance it is key that the learned latent representation exhibits critical fluctuations and scale-dependent flows consistent with the renormalization-group structure of percolation.
END MATTER
The scale-shared encoder–decoder applies the same local rules across lattice scales, compressing configurations to a latent vector for percolation prediction and reconstructing largest-cluster masks. With C = 32 and hidden width 64, the architecture has 55,189 trainable parameters independent of lattice size.
- Architectural details: The same encoder rule fθ and decoder rule f′θ are applied at every scale, so parameter count is independent of lattice size L.Scale sharing is used throughout the encoder–decoder architecture.
- Encoder: C = 32 channels and hidden width 64 define the encoder, whose 2×2 coarse-graining reduces sites by 4 and halves linear system size at each application.After k = log2 L iterations, the configuration maps to a single latent vector hk ∈ R^C.
- Encoder: The encoder readout predicts percolation probability from the single latent vector and is trained with binary cross-entropy on the percolation label.The latent representation is produced after iterating the shared coarse-graining rule across scales.
- Decoder: The decoder upsamples coarse representations, concatenates same-resolution encoder features, and applies a shared local 2×2 fine-graining rule with U-Net-like skip connections.The final decoder combines its output with the original binary configuration to predict largest-cluster membership probabilities.
- Parameter count: 55,189 trainable parameters comprise the full encoder–decoder, and this number is the same for all lattice sizes.For C = 32 and hidden width 64, the encoder rule contains 19,584 parameters, the classifier readout 2,177, the decoder rule 32,896, and the final decoder readout 532.